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We consider the equations of Navier-Stokes modeling viscous fluid flow past amoving or rotating obstacle in $\mathbb{R}^d$ subject to a prescribed velocitycondition at infinity. In contrast to previously known results, where theprescribed velocity vector is assumed to be parallel to the axis of rotation,in this paper we are interested in a general outflow velocity. In order to use$L^p$-techniques we introduce a new coordinate system, in which we obtain anon-autonomous partial differential equation with an unbounded drift term. Weprove that the linearized problem in $\mathbb{R}^d$ is solved by an evolutionsystem on $L^p_{\sigma}(\mathbb{R}^d)$ for $1<p<\infty$. For this we useresults about time-dependent Ornstein-Uhlenbeck operators. Finally, we prove,for $p\geq d$ and initial data $u_0\in L^p_{\sigma}(\mathbb{R}^d)$, theexistence of a unique mild solution to the full Navier-Stokes system.

On the Navier-Stokes equations with rotating effect and prescribed outflow velocity